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A228202 Number T(n,k,r) of partitions of an n X k X r rectangular cuboid into integer-sided cubes, considering only the list of parts; irregular triangle T(n,k,r), n >= k >= r >= 1 read by rows.

%I A228202 #32 Oct 27 2023 22:05:40
%S A228202 1,1,1,2,1,1,2,1,2,3,1,1,3,1,3,4,1,5,6,11,1,1,3,1,3,5,1,5,8,15,1,5,9,
%T A228202 17,19,1,1,4,1,4,7,1,7,11,24,1,7,16,34,40,1,10,23,52,80,121
%N A228202 Number T(n,k,r) of partitions of an n X k X r rectangular cuboid into integer-sided cubes, considering only the list of parts; irregular triangle T(n,k,r), n >= k >= r >= 1 read by rows.
%H A228202 Alois P. Heinz, <a href="/A228202/b228202.txt">Rows n = 1..6, flattened</a> (rows n=1..5 from Christopher Hunt Gribble)
%H A228202 Christopher Hunt Gribble, <a href="/A228202/a228202.cpp.txt">C++ program</a>
%e A228202 The irregular triangle begins:
%e A228202    r  1   2   3   4 ...
%e A228202 n,k
%e A228202 1,1   1
%e A228202 2,1   1
%e A228202 2,2   1   2
%e A228202 3,1   1
%e A228202 3,2   1   2
%e A228202 3,3   1   2   3
%e A228202 4,1   1
%e A228202 4,2   1   3
%e A228202 4,3   1   3   4
%e A228202 4,4   1   5   6  11
%e A228202 5,1   1
%e A228202 5,2   1   3
%e A228202 5,3   1   3   5
%e A228202 5,4   1   5   8  15
%e A228202 5,5   1   5   9  17  19
%e A228202 ...
%e A228202 T(4,4,3) = 6 because there are 6 partitions of a 4 X 4 X 3 rectangular cuboid into integer-sided cubes.  The partitions are:
%e A228202 48 1 X 1 X 1 cubes,
%e A228202 40 1 X 1 X 1 cubes and 1 2 X 2 X 2 cube,
%e A228202 32 1 X 1 X 1 cubes and 2 2 X 2 X 2 cubes,
%e A228202 24 1 X 1 X 1 cubes and 3 2 X 2 X 2 cubes,
%e A228202 16 1 X 1 X 1 cubes and 4 2 X 2 X 2 cubes,
%e A228202 21 1 X 1 X 1 cubes and 1 3 X 3 X 3 cube.
%Y A228202 T(n,n,n) = A129668(n).
%Y A228202 Cf. A224697.
%K A228202 nonn,tabf
%O A228202 1,4
%A A228202 _Christopher Hunt Gribble_, Aug 17 2013
%E A228202 21 more terms (row 6) from _Alois P. Heinz_, Aug 18 2013